1. Field of the Invention
This invention is concerned with a method for processing seismic data in the presence of steep dips and a laterally-varying wavefield propagation velocity.
2. Discussion of the Prior Art
In seismic reflection exploration, an energy source at or near the surface of the earth generates a wavefield in the earth. The wavefield radiates into the earth in all directions from the source. In its passage through earth formations, the wavefield is reflected from various discontinuities whence it is reflected back to the surface. Seismic transducers are disposed at or near the surface of the earth along a line of survey at each of a plurality of stations. The transducers detect the ground motion due to the various reflected wavefields. The detected ground motions are converted to electrical signals. The electrical signals are recorded, processed and displayed as a suite of time scale recordings generally termed a seismic section or seismic profile. A properly-processed seismic section images the attitude of earth formations along the line of survey.
In its most elementary form, the Z axis of a seismic section represents the two-way travel time of a wavefield from the source to the respective reflectors and back to the surface. The X axis represents the transducer locations or stations along the line of survey. It is customary, by means well-known to the art, to arrange the received seismic data in the form of zero-offset Common Midpoint (CMP) gathers. That is, the source points and the respective transducers are represented as being co-located at each station.
On the elementary seismic section, the travel times of reflected seismic events are displayed along the vertical Z axis directly beneath the respective CMP locations. In the case where a reflecting formation is tilted, that is, where it dips relative to the surface, the elementary seismic section is distorted for this reason: As we know from Mr. Snell's law, the angle of incidence must equal the angle of reflection. But the wavefield was beamed from and received at the same CMP location. Therefore, the incident point of the beam at the dipping reflector cannot lie directly beneath a selected CMP location; it must lie somewhere in space to one side of a selected station. The process of correcting the elementary seismic profile to correctly reposition the raw reflection data is called dip migration.
In the discussions to follow, the unqualified term "velocity" will be used as a short-hand expression that means the velocity of propagation of an acoustic wavefield through earth formations. Generally, for purposes of this discussion, the term is meant to apply to the propagation velocity of compressional waves although shear waves are not excluded. It is evident that the depth to a reflector is equal to the average velocity multiplied by one-half the travel time to that reflector, after allowance for dip angle.
The migration velocity is determined from the velocity spectrum generated from the seismic reflection-time data. The migration velocity is not necessarily the same as the formation velocity for dipping seismic reflection events.
The term "time", unless otherwise defined, means the two-way travel time of a wavefield from a source to a reflector, back to a receiver (transducer).
A number of well-known algorithms exist for time migration of seismic dips in a single pass. Exemplary known algorithms include the Stolt f-k migration, Kirchhoff-summation migration and finite difference migration. For relatively gentle dips up to about 45.degree. and a laterally invariant velocity, the known algorithms are reasonably accurate and computationally efficient. On the other hand, a certain algorithm, such as Stolt f-k migration, that can image steep dips approaching 90.degree., breaks down in the presence of a significant lateral velocity gradient. Each of the known methods has both strengths and weaknesses.
Cascaded migration is a technique that has been developed to handle steep dips as well as varying velocities. It involves iteratively migrating the time-dipping reflection data over a plurality of stages. See, for example, U.S. Pat. No. 4,745,585, issued 05/17/88 and U.S. Pat. 4,888,742, issued Dec. 19, 1989, both assigned to the assignee of this invention. The migration velocity for each stage becomes progressively lower than the true migration velocity. Because the migration velocity is relatively low for each stage, the perceived dip is relatively small and hence more accurate. The cumulative dips approach the true dip, stepwise, after n migration stages.
The migration process is applied over a series of travel-time windows of limited width. The width of a time window, sometimes referred to as a tau-step, is determined on the basis of the dip ranges to be examined and the areal velocity pattern, as a function of both formation depth and lateral velocity distribution. In cascaded migration, the migration velocities for each migration stage, for a given tau-step, are held constant and are always less than the true migration velocity. Cascaded migration can deliver accurate results, but the complexity of the multiple-stage approach is a hinderance, particularly when extended to three-dimensional (3-D) migration.
Residual migration is a two-step approach to migration. The migration is initially done using a velocity V.sub.1 that is lower than the correct migration velocity V. The migration is completed by applying a second, or residual, migration to the previously migrated data using a residual migration velocity V.sub.R, defined as a function of migrated time T by EQU V.sub.R.sup.2 (T)=V.sup.2 (T)-V.sub.1.sup.2 (T).
In this formulation, it is required that V.sub.1 be strictly constant. See, for example, Rothman et al. (1985) Residual Migration: Application and Limitations, Geophsics, v. 50, n. 1, pp 110-126.
In theory, a substantial portion of the migration can be done in the first stage, leaving a small amount of migration for the residual step. However, in practice, the limitation of using a constant velocity in the first stage often results in a residual migration velocity that will be quite high and not substantially reduced from the true migration velocity. A less restrictive but more complicated condition on the initial velocity allows the velocity to vary down to some fixed reflection travel-time T.sub.c, but still requires a constant velocity below that level. Also, above T.sub.c, the migration must have been completed in the first stage. Those requirements are based on the observation that unless the first migration velocity is constant below T.sub.c, the velocity used in the first stage of migration is influenced by velocities from layers deeper than the migrated position and is typically too high.
This invention provides a way to overcome that limitation by use of a dip-dependent residual velocity.